COMMUNICATION TUEORY OF SECIULCY SYSTEMS 711 



Both of these may be simple operations on their arguments without the 

 third equation 



A' = h{M, E) 



being simple. 



We may also point out that in investigating a new type of ciphering sys- 

 tem one of the best methods of attack is to consider how the key could be 

 determined if a sulllcient amount of M and E were given. 



The principle of confusion can be (and must be) used to create diOiculties 

 for the cr>'ptanalyst using probable word techniques. Given (or assuming) 

 M = mi , m-i , • ■ • , nis and E = ei , 62 , • ■ • , Cs the cryptanalyst can set up 

 equations for the different key elements k] , ko , ■ ■ ■ , kr (namely the en- 

 ciphering equations). 



ei = fi(mi ,m-i,---, nis \ki , ■■■ , kr) 

 €2 = Mtni , m-i , ■ ■ ■ , Ms ; ki , ■ • • , kr) 



es = fs(nii , nh , ■ ■ ■ , m, ; ki , ■ • • , kr) 



All is known, we assume, except the ki . Each of these equations should 

 therefore be complex in the ki , and involve many of them. Otherwise the 

 enemy can solve the simple ones and then the more complex ones by sub- 

 stitution. 



From the point of view of increasing confusion, it is desirable to have the 

 fi involve several rm , especially if these are not adjacent and hence less 

 correlated. This introduces the undesirable feature of error propagation, 

 however, for then each Ci will generally aflfect several w, in deciphering, and 

 an error will spread to all these. 



We conclude that much of the key should be used in an involved manner 

 in obtaining any cryptogram letter from the message to keep the work 

 characteristic high. Further a dependence on several uncorrelated w.- is 

 desirable, if some propagation of error can be tolerated. We are led by all 

 three of the arguments of these sections to consider "mi.xing transforma- 

 tions." 



25. Mixing Transformations 



A notion that has proved valuable in certain branches of probability 

 theory is the concept of a mixing transj'ormation. Suppose we have a prob- 

 ability or measure space fi and a measure preserving transformation F of 

 the space into itself, that is, a transformation such that the measure of a 



